Andrea Scozzari Università Niccolò Cusano Telematica, Roma

Transcription

1 Andrea Scozzari Università Niccolò Cusano Telematica, Roma Joint research with: J. Puerto Universidad de Sevilla F. Ricca Sapienza, Università di Roma The Exploratory Workshop on Locational Analysis: Trends on Theory and Applications Sevilla, November

2 Scope: -To update (some of) the results presented in the most recent literature surveys on the location of extensive facilities. (S.L. Hakimi, E.F. Schmeichel, M. Labbè, Networks, 1993, J.A. Mesa, T. B. Boffey, EJOR, 1996) -To give an overview of (possible) new lines of research on the location of extensive facilities on Networks. Extensive Facilities are activities that are too large to be modelled as points: public transit lines, pipilines, emergency routes high speed communication networks Paths Trees

3 Given a weighted graph G=(V,E,w,l) the problem consists of locating one path or one tree minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. Median criterion D ( w d( v, v V v Center criterion E( max w d( v, v V v

4 Given a weighted graph G=(V,E,w,l) the problem consists of locating one path or one tree minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. Paths or Trees are classified w.r.t. the following criteria: - discrete: the tips of the facility are all nodes of G; - continuous: at least one tip of the facility belongs to the interior of an edge. In this talk we are mostly interested on path location problems

5 Given a weighted Graph G=(V,E,w,l) the problem consists of locating one path minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. Negative Results Locating a continuous/discrete path minimizing the Maximum Distance or the Sum of the Distances with/without a length constraint is NP- Complete on: - Planar Cubic Graphs (reduction from the Hamiltonian path problem S.L. Hakimi, E.F. Schmeichel, M. Labbè, Networks, 1993) - Bipartite Graphs (reduction from the Hamiltonian path problem, M.R. Garey, D.S. Johnson, 1979) - Grid Graphs (reduction from the Hamiltonian path problem, R.I. Becker, I. Lari, A. Scozzari, G. Storchi, Annals of OR, 2007)

6 Negative Results Locating a discrete/continuous path minimizing the Sum of the Distances with length at most L is NP-Complete on: -cactus graphs (reduction from Partition with disjoint pairs M.B. Richey, Networks, 1990, I. Lari, F. Ricca, A. Scozzari, EJOR, 2008) Finding a path such that the Sum of the Distances is exactly equal to K > 0 is NP-Complete on -cactus graphs (reduction from Partition with disjoint pairs I. Lari, F. Ricca, A. Scozzari, R.I. Becker, Networks, 2011)

7 Given a weighted graph G=(V,E,w,l) find one path minimizing the Sum of the Distances from the nodes of G to the facility without a length constraint. Special case: a connected outerplanar graph G=(V,E,w), with weights equal to 1 assigned to all the edges of G, and (arbitrary) nonnegative weights associated to its nodes. (I. Lari, F. Ricca, A. Scozzari, R.I. Becker, Networks, 2011) Example of a connected outerplanar graph

8 -Given a connected outerplanar graph G=(V,E,w), with weights equal to 1 assigned to all the edges of G, and (arbitrary) nonnegative weights associated to its nodes, find one path minimizing the Sum of the Distances from the nodes of G to the facility without a length constraint. O(kn) with k the number of connected components of G (I. Lari, F. Ricca, A. Scozzari, R.I. Becker, Networks, 2011) O(n) for a biconnected outerplanar graph G having fixed the endnodes of the path. (I. Lari, F. Ricca, A. Scozzari, R.I. Becker, Networks, 2011) s t

9 Negative Results Locating a discrete/continuous path minimizing the Maximum Distance with length exactly equal to L is NP-Complete on: Outerplanar graphs (red. from Partition with Disjoint Pairs M.B. Richey, Networks, 1990) Finding a discrete path minimizing the Maximum Distance with length at most L is: O( n 12 logb) Polynomially solvable on Series-Parallel graphs (M.B. Richey, Networks, 1990) The same algorithm can be applied for the median criterion with a pseudo-polynomial time complexity.

10 Possible directions Starting from: (I. Lari, F. Ricca, A. Scozzari, R.I. Becker, Networks, 2011) Starting from: M.B. Richey, Networks, Generalize to the case of edge weighted outerplanar graphs; - Consider the median path problems without the length constraint on Series-Parallel graphs and/or on more general classes of graphs. - Introduce the length constraint and study the problem on special classes of graphs characterized by hereditary properties.

11 Positive Results Given a weighted tree T=(V,E,w,l) find one path minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. Finding a discrete path minimizing the Sum of the Distances with length at most L is solvable in: O(n log n) (S. Alstrup, P.W. Lauridsen, P. Sommerlund, M. Thorup, Lecture Notes in Comp. Sci., Vol. 1272, 1997) Ω(n log n) (B-F. Wang, T-C. Lin, C-H. Lin, S-C. Ku, Information and Computation, 2008) even if w(v)=1 v V.

12 Positive Results Given a weighted tree T=(V,E,w,l) find one path minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. Finding a discrete path minimizing the Sum of the Distances with length at most L is solvable in: O(n log n) (S. Alstrup, P.W. Lauridsen, P. Sommerlund, M. Thorup, Lecture Notes in Comp. Sci., Vol. 1272, 1997) Ω(n log n) (B-F. Wang, T-C. Lin, C-H. Lin, S-C. Ku, Information and Computation, 2008) even if w(v)=1 v V.

13 Positive Results Given a weighted tree T=(V,E,w,l) find one path minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) distance from the nodes of G to the facility. Finding a continuous path minimizing the Sum of the Distances with length at most L is solvable in: O(n log(n) (n)) (S. Alstrup, P.W. Lauridsen, P. Sommerlund, M. Thorup, Lecture Notes in Comp. Sci., Vol. 1272, 1997) For this problem a lower bound of Ω(n log n) time is provided in (B-F. Wang, T-C. Lin, C-H. Lin, S-C. Ku, Information and Computation, 2008) but in the case of a conditional median path.

14 Positive Results Given a weighted tree T=(V,E,w,l) find one path minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. Theorem 1 (B. Bhattacharya, Y. Hu, Q. Shi, A. Tamir, Lecture Notes in Comp. Sci., Vol. 4288, 2006) The discrete/continuous weighted path center problem with/without length constraint in tree networks can be solved in linear time.

15 Positive Results Given a weighted tree T=(V,E,w,l) find one path minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. Theorem 1 (B. Bhattacharya, Y. Hu, Q. Shi, A. Tamir, Lecture Notes in Comp. Sci., Vol. 4288, 2006) The discrete/continuous weighted path center problem with/without length constraint in tree networks can be solved in linear time.

16 Possible directions

17 Given a weighted graph G=(V,E,w,l) find p 2 disjoint paths minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. Locating continuous/discrete p 2 disjoint paths minimizing the Maximum Distance or the Sum of the Distances with/without a length constraint is NP-Complete on arbitrary graphs (reduction from the Covering by p path problem, S.L. Hakimi, E.F. Schmeichel, M. Labbè, Networks, 1993) G u G. Reduction used for proving the NP-Completeness of Covering by p path problem p

18 Given a weighted graph G=(V,E,w,l) find p 2 disjoint paths minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. Locating continuous/discrete p 2 disjoint paths minimizing the Maximum Distance or the Sum of the Distances with/without a length constraint is NP-Complete on arbitrary graphs (reduction from the Covering by p path problem, S.L. Hakimi, E.F. Schmeichel, M. Labbè, Networks, 1993) G u G. Reduction used for proving the NP-Completeness of Covering by p path problem p

19 Possible directions Given a weighted graph G=(V,E,w,l) find p 2 disjoint paths minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) distance from the nodes of G to the facility. Since the covering by p-path problem has been shown to be NPcomplete by a reduction from the hamiltonian path problem we have: Locating continuous/discrete p 2 disjoint paths minimizing the Maximum Distance or the Sum of the Distances with/without a length constraint is NP-Complete also on Planar Cubic Graphs Bipartite graphs Grid Graphs

20 Given a weighted Tree T=(V,E,w,l) find p 2 disjoint paths minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. - p is an input variable: Locate p disjoint discrete paths minimizing the Sum of the Distances with cost at most L is NP-Complete on trees (S.L. Hakimi, E.F. Schmeichel, M. Labbè, Networks, 1993). Locate p disjoint continuous paths minimizing the Sum of the Distances with cost at most L OPEN!! (A. Tamir, T.J. Lowe Networks, 1992). - p fixed: the problems are polynomially solvable (S.L. Hakimi, E.F. Schmeichel, M. Labbè, Networks, 1993).

21 Given a weighted Tree T=(V,E,w,l) find p 2 disjoint paths minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. - p=2 Locate 2 disjoint discrete paths minimizing the Sum of the Distances without length contraints: O(n) (B-F. Wang, J-J. Lin, Lecture Notes in Comp. Sci, Vol. 1969, 2000). - p>2 and Fixed O(n p-1 ) (B-F. Wang, J-J. Lin, Lecture Notes in Comp. Sci, Vol. 1969, 2000).

22 Given a weighted Tree T=(V,E,w,l) find p 2 disjoint paths minimizing: - the Sum of the (weighted) Distances from the nodes of G to the facility; - the Maximum (weighted) Distance from the nodes of G to the facility. - p is an input variable: Locate p 2 disjoint discrete/continuous paths minimizing the Maximum Distances is polynomially solvable on trees in O(n 3 p 2 ) (A. Tamir, T.J. Lowe, Networks, 1992).

23 Possible directions

24 Given a weighted graph G=(V,E,w,l) find one path minimizing: - some Equity measures; - Measures that take into account the uncertainty of some problem s parameters.

25 Given a weighted tree T=(V,E,w,l) find one path minimizing: Some Equity measures. - Locating a path on T with objective given by a convex combination of the weighted center and weighted median functions. O(n log n) (J. Puerto, A.M., Rodriguez-Chia, A. Tamir, D. Perez-Brito, Networks, 2006)

26 Given a weighted tree T=(V,E,w,l) find one path minimizing: Some Equity measures. (J. Puerto, F. Ricca, A. Scozzari, Discrete Applied Math., 2009) Equity refers to the distribution of the clients' demand in a geographical area and the objective is to locate facilities in order to ensure a low variability of the distribution of the distances from the demand points (clients) to a facility. The weighted Range objective function is defined as follows: R( max [ w u V \ P u d( u, ] min [ w u V \ P u d( u, ]

27 Given a weighted tree T=(V,E,w,l) find one path minimizing: Some Equity measures. (J. Puerto, F. Ricca, A. Scozzari, Discrete Applied Math., 2009) The problem of locating a discrete/continuous path minimizing the Range objective function with/without a length constraint is NP- Complete on: - Planar Cubic Graphs - Bipartite graphs - Grid Graphs Since it contains as special case the (weighted) path center problem (J. Puerto, F. Ricca, A. Scozzari, Discrete Applied Math., 2009)

28 Given a weighted tree T=(V,E,w,l) find one path minimizing: Some Equity measures. (J. Puerto, F. Ricca, A. Scozzari, Discrete Applied Math., 2009) Find a discrete path P minimizing the Range with length L( Sss E( max [ w u V \ P R( Sss 1.1 u d( u, ] Sss E( Sss R( max [ w u V \ P 0.9 u d( u, ] 2

29 Given a weighted tree T=(V,E,w,l) find one path minimizing: Some Equity measures. Problem min R( UNWEIGTHED (*) WEIGTHED (**) Without length constraint Without length constraints With length constraints Discrete Continuous Discrete Continuous Discrete Continuous P1 O(n) O(n 2 ) O(n 2 ) O(n 3 ) O(n 2 ) O(n 3 ) min E( P2 O(n) O(n) O(n 2 ) O(n 2 ) O(n 2 ) O(n 3 ) s. t ( max ( P3 O(n) O(n) O(n 2 ) O(n 2 ) O(n 2 ) O(n 3 ) s. t E( (*) J. Puerto, F. Ricca, A. Scozzari, Discrete Applied Math., 2009 (**) J. Puerto, F. Ricca, A. Scozzari, CTW 2011, submitted to Discrete Applied Math.

30 Problem min R( Future work UNWEIGTHED Without length constraint - No results about Nestedness properties. - For the weighted case, can we exploit a suitable decomposition of T? - Find a Tree minimizing the Range. Without length constraints - Extend the problem on more general classes of graph. WEIGTHED With length constraints Discrete Continuous Discrete Continuous Discrete Continuous P1 O(n) O(n 2 ) O(n 2 ) O(n 3 ) O(n 2 ) O(n 3 ) min E( P2 O(n) O(n) O(n 2 ) O(n 2 ) O(n 2 ) O(n 3 ) s. t ( max ( P3 O(n) O(n) O(n 2 ) O(n 2 ) O(n 2 ) O(n 3 ) s. t E(

31 Given a weighted tree T=(V,E,w,l) find one path minimizing: Some Equity measures. (J. Puerto, F. Ricca, A. Scozzari, Networks, 2009) Find a discrete/continuous defined as: path P minimizing the variance objective function Var( v V w v ( d( v, D( ) 2 where D ( w d( v, v V v

32 Given a weighted tree T=(V,E,w,l) find one path minimizing: Some Equity measures. Find a discrete/continuous without length constraint: path P minimizing the variance objective function O(n 2 log n) (T. Caceres, M.C. Lopez-de-los-Mozos, J.A. Mesa, Discrete Applied Math., 2004) (J. Puerto, F. Ricca, A. Scozzari, Networks, 2009) Problem Without length constraints With length constraints Discrete Continuous Discrete Continuous Var( O(n 2 ) O(n 2 ) O(n 2 ) O(n 2 )

33 Given a weighted tree T=(V,E,w,l) find one path minimizing: Some Equity measures. (J. Puerto, F. Ricca, A. Scozzari, Networks, 2009) The idea is to exploit the well-known variance decomposition formula to obtain efficient saving functions. Var( v V w d( v, v 2 D( 2 - In the unconstrained version of the problem the optimal discrete or continuous path does not satisfy any Nestedness properties. - A discrete unconstrained path P that minimizes the variance function does not necessarily have its end vertices at the leaves of T.

34 Given a weighted tree T=(V,E,w,l) find one path minimizing: Some Equity measures. (J. Puerto, F. Ricca, A. Scozzari, Networks, 2009) Var( v V w d( v, v 2 D( 2 Future work - No results about the computational complexity of the problem on arbitrary graphs or on special classes of graphs. - Can we exploit a suitable decomposition of T? - Find a Tree minimizing the variance function.

35 REGRET PROBLEMS Complete uncertainty on some parameters of the problem (e.g., resource limitations, vertex or edge weights, etc.). uncertain parameters are given as intervals of values. Scenario Each possible choice of values for the unknown parameter. Objective To minimize the worst-case opportunity loss in the objective function.

36 PATH REGRET PROBLEMS W Ω be a given scenario P a path in a network F(W, any given evaluation function for P under W. REGRET R(W, = F(W, - F(W,Q*(W)) where Q*(W) is the optimal path under scenario W. The regret R(W, measures the deviation from the optimal objective function value when P is chosen instead of Q*(W).

37 PATH REGRET PROBLEMS the maximum regret of P is R( = max W Ω R(W, The aim of the minimax regret approach is to minimize the worst case deviation given by the maximum regret. OBJECTIVE FUNCTION min P max W Ω R(W,

38 PATH REGRET PROBLEMS ON TREES Given a tree T, with fixed edge weights and uncertain vertex weights, find a path P in T that minimizes R(. (J. Puerto, F. Ricca, A. Scozzari, Networks, 2011) CENTER MEDIAN CENT-DIAN

39 EXAMPLE

40 RESULTS PROBLEM COMPLEXITY Regret PATH CENTER O(n 2 ) Regret PATH MEDIAN O(n 4 ) Regret PATH CENT-DIAN O(n 5 log n) J. Puerto, F. Ricca, A. Scozzari, Minimax Regret Path Location on Trees Networks 58 (2011), pp

41 OPEN PROBLEMS Studying REGRET PROBLEMS: Location of a path with uncertainty only on the edge weights, or on both edge and vertex weights. Location of a path with an additional constraint on the length of the path. Location of a subtree with a bound on its total length (uncertain weights only for the vertices). Location of a subtree with a bound on its diameter (uncertain weights only for the vertices).